In a manifold with non-trivial fundamental group, generalized linking numbers and Arf invariants can be defined by examining singularities in generic homotopies of submanifolds. For knots in a 3-manifold M, these invariants depend on intersections among disks, spheres and tori, and have clear geometric characterizations. For instance, a link in M has vanishing generalized Arf invariant iff it bounds immersed disks in M cross I which are homotopic to disjointly embedded disks after connected-summing with copies of S^2 cross S^2.